Data.Profunctor.Strong

Strength

class Profunctor p => Strong p where

Generalizing Star of a strong Functor

Note: Every Functor in Haskell is strong with respect to (,).

This describes profunctor strength with respect to the product structure of Hask.

http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf

Minimal complete definition

first' | second'

Methods

first' :: p a b -> p (a, c) (b, c)

Laws:

first' ≡ dimap swap swap . second'
lmap fst ≡ rmap fst . first'
lmap (second' f) . first' ≡ rmap (second' f) . first'
first' . first' ≡ dimap assoc unassoc . first' where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

second' :: p a b -> p (c, a) (c, b)

Laws:

second' ≡ dimap swap swap . first'
lmap snd ≡ rmap snd . second'
lmap (first' f) . second' ≡ rmap (first' f) . second'
second' . second' ≡ dimap unassoc assoc . second' where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

Instances

Monad m => Strong (Kleisli m)

Defined in Data.Profunctor.Strong

Methods

first' :: Kleisli m a b -> Kleisli m (a, c) (b, c)

second' :: Kleisli m a b -> Kleisli m (c, a) (c, b)

Strong (Pastro p)

Defined in Data.Profunctor.Strong

Methods

first' :: Pastro p a b -> Pastro p (a, c) (b, c)

second' :: Pastro p a b -> Pastro p (c, a) (c, b)

Profunctor p => Strong (Tambara p)

Defined in Data.Profunctor.Strong

Methods

first' :: Tambara p a b -> Tambara p (a, c) (b, c)

second' :: Tambara p a b -> Tambara p (c, a) (c, b)

Strong p => Strong (Closure p)

Defined in Data.Profunctor.Closed

Methods

first' :: Closure p a b -> Closure p (a, c) (b, c)

second' :: Closure p a b -> Closure p (c, a) (c, b)

Strong (FreeTraversing p)

Defined in Data.Profunctor.Traversing

Methods

first' :: FreeTraversing p a b -> FreeTraversing p (a, c) (b, c)

second' :: FreeTraversing p a b -> FreeTraversing p (c, a) (c, b)

Profunctor p => Strong (CofreeTraversing p)

Defined in Data.Profunctor.Traversing

Methods

first' :: CofreeTraversing p a b -> CofreeTraversing p (a, c) (b, c)

second' :: CofreeTraversing p a b -> CofreeTraversing p (c, a) (c, b)

Strong (FreeMapping p)

Defined in Data.Profunctor.Mapping

Methods

first' :: FreeMapping p a b -> FreeMapping p (a, c) (b, c)

second' :: FreeMapping p a b -> FreeMapping p (c, a) (c, b)

Profunctor p => Strong (CofreeMapping p)

Defined in Data.Profunctor.Mapping

Methods

first' :: CofreeMapping p a b -> CofreeMapping p (a, c) (b, c)

second' :: CofreeMapping p a b -> CofreeMapping p (c, a) (c, b)

Strong p => Strong (Coyoneda p)

Defined in Data.Profunctor.Yoneda

Methods

first' :: Coyoneda p a b -> Coyoneda p (a, c) (b, c)

second' :: Coyoneda p a b -> Coyoneda p (c, a) (c, b)

Strong p => Strong (Yoneda p)

Defined in Data.Profunctor.Yoneda

Methods

first' :: Yoneda p a b -> Yoneda p (a, c) (b, c)

second' :: Yoneda p a b -> Yoneda p (c, a) (c, b)

Strong ((->) :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

first' :: (a -> b) -> (a, c) -> (b, c)

second' :: (a -> b) -> (c, a) -> (c, b)

Strong (Forget r :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

first' :: Forget r a b -> Forget r (a, c) (b, c)

second' :: Forget r a b -> Forget r (c, a) (c, b)

Functor m => Strong (Star m)

Defined in Data.Profunctor.Strong

Methods

first' :: Star m a b -> Star m (a, c) (b, c)

second' :: Star m a b -> Star m (c, a) (c, b)

Contravariant f => Strong (Clown f :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

first' :: Clown f a b -> Clown f (a, c) (b, c)

second' :: Clown f a b -> Clown f (c, a) (c, b)

Arrow p => Strong (WrappedArrow p)

Arrow is Strong Category

Defined in Data.Profunctor.Strong

Methods

first' :: WrappedArrow p a b -> WrappedArrow p (a, c) (b, c)

second' :: WrappedArrow p a b -> WrappedArrow p (c, a) (c, b)

(Strong p, Strong q) => Strong (Sum p q)

Defined in Data.Profunctor.Strong

Methods

first' :: Sum p q a b -> Sum p q (a, c) (b, c)

second' :: Sum p q a b -> Sum p q (c, a) (c, b)

(Strong p, Strong q) => Strong (Product p q)

Defined in Data.Profunctor.Strong

Methods

first' :: Product p q a b -> Product p q (a, c) (b, c)

second' :: Product p q a b -> Product p q (c, a) (c, b)

(Functor f, Strong p) => Strong (Tannen f p)

Defined in Data.Profunctor.Strong

Methods

first' :: Tannen f p a b -> Tannen f p (a, c) (b, c)

second' :: Tannen f p a b -> Tannen f p (c, a) (c, b)

(Strong p, Strong q) => Strong (Procompose p q)

Defined in Data.Profunctor.Composition

Methods

first' :: Procompose p q a b -> Procompose p q (a, c) (b, c)

second' :: Procompose p q a b -> Procompose p q (c, a) (c, b)

(Functor f, Strong p) => Strong (Cayley f p)

Defined in Data.Profunctor.Cayley

Methods

first' :: Cayley f p a b -> Cayley f p (a, c) (b, c)

second' :: Cayley f p a b -> Cayley f p (c, a) (c, b)

uncurry' :: Strong p => p a (b -> c) -> p (a, b) c

strong :: Strong p => (a -> b -> c) -> p a b -> p a c

newtype Tambara p a b

Tambara cofreely makes any Profunctor Strong.

Constructors

Tambara

Fields

• runTambara :: forall c. p (a, c) (b, c)

Instances

ProfunctorComonad Tambara

Defined in Data.Profunctor.Strong

Methods

proextract :: Profunctor p => Tambara p :-> p

produplicate :: Profunctor p => Tambara p :-> Tambara (Tambara p)

ProfunctorAdjunction Pastro Tambara

Defined in Data.Profunctor.Strong

Methods

unit :: Profunctor p => p :-> Tambara (Pastro p)

counit :: Profunctor p => Pastro (Tambara p) :-> p

Arrow p => Arrow (Tambara p)

Defined in Data.Profunctor.Strong

Methods

arr :: (b -> c) -> Tambara p b c

first :: Tambara p b c -> Tambara p (b, d) (c, d)

second :: Tambara p b c -> Tambara p (d, b) (d, c)

(***) :: Tambara p b c -> Tambara p b' c' -> Tambara p (b, b') (c, c')

(&&&) :: Tambara p b c -> Tambara p b c' -> Tambara p b (c, c')

ArrowZero p => ArrowZero (Tambara p)

Defined in Data.Profunctor.Strong

Methods

zeroArrow :: Tambara p b c

ArrowPlus p => ArrowPlus (Tambara p)

Defined in Data.Profunctor.Strong

Methods

(<+>) :: Tambara p b c -> Tambara p b c -> Tambara p b c

ArrowChoice p => ArrowChoice (Tambara p)

Defined in Data.Profunctor.Strong

Methods

left :: Tambara p b c -> Tambara p (Either b d) (Either c d)

right :: Tambara p b c -> Tambara p (Either d b) (Either d c)

(+++) :: Tambara p b c -> Tambara p b' c' -> Tambara p (Either b b') (Either c c')

(|||) :: Tambara p b d -> Tambara p c d -> Tambara p (Either b c) d

ArrowApply p => ArrowApply (Tambara p)

Defined in Data.Profunctor.Strong

Methods

app :: Tambara p (Tambara p b c, b) c

ArrowLoop p => ArrowLoop (Tambara p)

Defined in Data.Profunctor.Strong

Methods

loop :: Tambara p (b, d) (c, d) -> Tambara p b c

Profunctor p => Profunctor (Tambara p)

Defined in Data.Profunctor.Strong

Methods

dimap :: (a -> b) -> (c -> d) -> Tambara p b c -> Tambara p a d

lmap :: (a -> b) -> Tambara p b c -> Tambara p a c

rmap :: (b -> c) -> Tambara p a b -> Tambara p a c

(#.) :: Coercible c b => q b c -> Tambara p a b -> Tambara p a c

(.#) :: Coercible b a => Tambara p b c -> q a b -> Tambara p a c

Profunctor p => Strong (Tambara p)

Defined in Data.Profunctor.Strong

Methods

first' :: Tambara p a b -> Tambara p (a, c) (b, c)

second' :: Tambara p a b -> Tambara p (c, a) (c, b)

Choice p => Choice (Tambara p)

Defined in Data.Profunctor.Choice

Methods

left' :: Tambara p a b -> Tambara p (Either a c) (Either b c)

right' :: Tambara p a b -> Tambara p (Either c a) (Either c b)

Category p => Category (Tambara p :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

id :: Tambara p a a

(.) :: Tambara p b c -> Tambara p a b -> Tambara p a c

ProfunctorFunctor Tambara

Defined in Data.Profunctor.Strong

Methods

promap :: Profunctor p => (p :-> q) -> Tambara p :-> Tambara q

Profunctor p => Functor (Tambara p a)

Defined in Data.Profunctor.Strong

Methods

fmap :: (a0 -> b) -> Tambara p a a0 -> Tambara p a b

(<$) :: a0 -> Tambara p a b -> Tambara p a a0

(Profunctor p, Arrow p) => Applicative (Tambara p a)

Defined in Data.Profunctor.Strong

Methods

pure :: a0 -> Tambara p a a0

(<*>) :: Tambara p a (a0 -> b) -> Tambara p a a0 -> Tambara p a b

liftA2 :: (a0 -> b -> c) -> Tambara p a a0 -> Tambara p a b -> Tambara p a c

(*>) :: Tambara p a a0 -> Tambara p a b -> Tambara p a b

(<*) :: Tambara p a a0 -> Tambara p a b -> Tambara p a a0

(Profunctor p, ArrowPlus p) => Alternative (Tambara p a)

Defined in Data.Profunctor.Strong

Methods

empty :: Tambara p a a0

(<|>) :: Tambara p a a0 -> Tambara p a a0 -> Tambara p a a0

some :: Tambara p a a0 -> Tambara p a [a0]

many :: Tambara p a a0 -> Tambara p a [a0]

ArrowPlus p => Semigroup (Tambara p a b)

Defined in Data.Profunctor.Strong

Methods

(<>) :: Tambara p a b -> Tambara p a b -> Tambara p a b

sconcat :: NonEmpty (Tambara p a b) -> Tambara p a b

stimes :: Integral b0 => b0 -> Tambara p a b -> Tambara p a b

ArrowPlus p => Monoid (Tambara p a b)

Defined in Data.Profunctor.Strong

Methods

mempty :: Tambara p a b

mappend :: Tambara p a b -> Tambara p a b -> Tambara p a b

mconcat :: [Tambara p a b] -> Tambara p a b

tambara :: Strong p => (p :-> q) -> p :-> Tambara q

tambara (untambara f) ≡ f
untambara (tambara f) ≡ f

untambara :: Profunctor q => (p :-> Tambara q) -> p :-> q

tambara (untambara f) ≡ f
untambara (tambara f) ≡ f

data Pastro p a b where

Pastro -| Tambara

Pastro p ~ exists z. Costar ((,)z) Procompose p Procompose Star ((,)z)

Pastro freely makes any Profunctor Strong.

Constructors

Pastro :: ((y, z) -> b) -> p x y -> (a -> (x, z)) -> Pastro p a b

Instances

ProfunctorMonad Pastro

Defined in Data.Profunctor.Strong

Methods

proreturn :: Profunctor p => p :-> Pastro p

projoin :: Profunctor p => Pastro (Pastro p) :-> Pastro p

ProfunctorAdjunction Pastro Tambara

Defined in Data.Profunctor.Strong

Methods

unit :: Profunctor p => p :-> Tambara (Pastro p)

counit :: Profunctor p => Pastro (Tambara p) :-> p

Profunctor (Pastro p)

Defined in Data.Profunctor.Strong

Methods

dimap :: (a -> b) -> (c -> d) -> Pastro p b c -> Pastro p a d

lmap :: (a -> b) -> Pastro p b c -> Pastro p a c

rmap :: (b -> c) -> Pastro p a b -> Pastro p a c

(#.) :: Coercible c b => q b c -> Pastro p a b -> Pastro p a c

(.#) :: Coercible b a => Pastro p b c -> q a b -> Pastro p a c

Strong (Pastro p)

Defined in Data.Profunctor.Strong

Methods

first' :: Pastro p a b -> Pastro p (a, c) (b, c)

second' :: Pastro p a b -> Pastro p (c, a) (c, b)

ProfunctorFunctor Pastro

Defined in Data.Profunctor.Strong

Methods

promap :: Profunctor p => (p :-> q) -> Pastro p :-> Pastro q

pastro :: Strong q => (p :-> q) -> Pastro p :-> q

pastro (unpastro f) ≡ f
unpastro (pastro f) ≡ f

unpastro :: (Pastro p :-> q) -> p :-> q

pastro (unpastro f) ≡ f
unpastro (pastro f) ≡ f

Costrength

class Profunctor p => Costrong p where

Analogous to ArrowLoop, loop = unfirst

Minimal complete definition

unfirst | unsecond

Methods

unfirst :: p (a, d) (b, d) -> p a b

Laws:

unfirst ≡ unsecond . dimap swap swap
lmap (,()) ≡ unfirst . rmap (,())
unfirst . lmap (second f) ≡ unfirst . rmap (second f)
unfirst . unfirst = unfirst . dimap assoc unassoc where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

unsecond :: p (d, a) (d, b) -> p a b

Laws:

unsecond ≡ unfirst . dimap swap swap
lmap ((),) ≡ unsecond . rmap ((),)
unsecond . lmap (first f) ≡ unsecond . rmap (first f)
unsecond . unsecond = unsecond . dimap unassoc assoc where
  assoc ((a,b),c) = (a,(b,c))
  unassoc (a,(b,c)) = ((a,b),c)

Instances

MonadFix m => Costrong (Kleisli m)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Kleisli m (a, d) (b, d) -> Kleisli m a b

unsecond :: Kleisli m (d, a) (d, b) -> Kleisli m a b

Costrong (Tagged :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Tagged (a, d) (b, d) -> Tagged a b

unsecond :: Tagged (d, a) (d, b) -> Tagged a b

Costrong (Copastro p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Copastro p (a, d) (b, d) -> Copastro p a b

unsecond :: Copastro p (d, a) (d, b) -> Copastro p a b

Costrong (Cotambara p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Cotambara p (a, d) (b, d) -> Cotambara p a b

unsecond :: Cotambara p (d, a) (d, b) -> Cotambara p a b

Costrong p => Costrong (Coyoneda p)

Defined in Data.Profunctor.Yoneda

Methods

unfirst :: Coyoneda p (a, d) (b, d) -> Coyoneda p a b

unsecond :: Coyoneda p (d, a) (d, b) -> Coyoneda p a b

Costrong p => Costrong (Yoneda p)

Defined in Data.Profunctor.Yoneda

Methods

unfirst :: Yoneda p (a, d) (b, d) -> Yoneda p a b

unsecond :: Yoneda p (d, a) (d, b) -> Yoneda p a b

Costrong ((->) :: Type -> Type -> Type)

Defined in Data.Profunctor.Strong

Methods

unfirst :: ((a, d) -> (b, d)) -> a -> b

unsecond :: ((d, a) -> (d, b)) -> a -> b

Functor f => Costrong (Cokleisli f)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Cokleisli f (a, d) (b, d) -> Cokleisli f a b

unsecond :: Cokleisli f (d, a) (d, b) -> Cokleisli f a b

Functor f => Costrong (Costar f)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Costar f (a, d) (b, d) -> Costar f a b

unsecond :: Costar f (d, a) (d, b) -> Costar f a b

ArrowLoop p => Costrong (WrappedArrow p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: WrappedArrow p (a, d) (b, d) -> WrappedArrow p a b

unsecond :: WrappedArrow p (d, a) (d, b) -> WrappedArrow p a b

(Costrong p, Costrong q) => Costrong (Sum p q)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Sum p q (a, d) (b, d) -> Sum p q a b

unsecond :: Sum p q (d, a) (d, b) -> Sum p q a b

(Costrong p, Costrong q) => Costrong (Product p q)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Product p q (a, d) (b, d) -> Product p q a b

unsecond :: Product p q (d, a) (d, b) -> Product p q a b

(Functor f, Costrong p) => Costrong (Tannen f p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Tannen f p (a, d) (b, d) -> Tannen f p a b

unsecond :: Tannen f p (d, a) (d, b) -> Tannen f p a b

(Corepresentable p, Corepresentable q) => Costrong (Procompose p q)

Defined in Data.Profunctor.Composition

Methods

unfirst :: Procompose p q (a, d) (b, d) -> Procompose p q a b

unsecond :: Procompose p q (d, a) (d, b) -> Procompose p q a b

(Functor f, Costrong p) => Costrong (Cayley f p)

Defined in Data.Profunctor.Cayley

Methods

unfirst :: Cayley f p (a, d) (b, d) -> Cayley f p a b

unsecond :: Cayley f p (d, a) (d, b) -> Cayley f p a b

data Cotambara q a b where

Cotambara cofreely constructs costrength

Constructors

Cotambara :: Costrong r => (r :-> q) -> r a b -> Cotambara q a b

Instances

ProfunctorComonad Cotambara

Defined in Data.Profunctor.Strong

Methods

proextract :: Profunctor p => Cotambara p :-> p

produplicate :: Profunctor p => Cotambara p :-> Cotambara (Cotambara p)

ProfunctorAdjunction Copastro Cotambara

Defined in Data.Profunctor.Strong

Methods

unit :: Profunctor p => p :-> Cotambara (Copastro p)

counit :: Profunctor p => Copastro (Cotambara p) :-> p

Profunctor (Cotambara p)

Defined in Data.Profunctor.Strong

Methods

dimap :: (a -> b) -> (c -> d) -> Cotambara p b c -> Cotambara p a d

lmap :: (a -> b) -> Cotambara p b c -> Cotambara p a c

rmap :: (b -> c) -> Cotambara p a b -> Cotambara p a c

(#.) :: Coercible c b => q b c -> Cotambara p a b -> Cotambara p a c

(.#) :: Coercible b a => Cotambara p b c -> q a b -> Cotambara p a c

Costrong (Cotambara p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Cotambara p (a, d) (b, d) -> Cotambara p a b

unsecond :: Cotambara p (d, a) (d, b) -> Cotambara p a b

ProfunctorFunctor Cotambara

Defined in Data.Profunctor.Strong

Methods

promap :: Profunctor p => (p :-> q) -> Cotambara p :-> Cotambara q

Functor (Cotambara p a)

Defined in Data.Profunctor.Strong

Methods

fmap :: (a0 -> b) -> Cotambara p a a0 -> Cotambara p a b

(<$) :: a0 -> Cotambara p a b -> Cotambara p a a0

cotambara :: Costrong p => (p :-> q) -> p :-> Cotambara q

cotambara . uncotambara ≡ id
uncotambara . cotambara ≡ id

uncotambara :: Profunctor q => (p :-> Cotambara q) -> p :-> q

cotambara . uncotambara ≡ id
uncotambara . cotambara ≡ id

newtype Copastro p a b

Copastro -| Cotambara

Copastro freely constructs costrength

Constructors

Copastro

Fields

• runCopastro :: forall r. Costrong r => (forall x y. p x y -> r x y) -> r a b

Instances

ProfunctorMonad Copastro

Defined in Data.Profunctor.Strong

Methods

proreturn :: Profunctor p => p :-> Copastro p

projoin :: Profunctor p => Copastro (Copastro p) :-> Copastro p

ProfunctorAdjunction Copastro Cotambara

Defined in Data.Profunctor.Strong

Methods

unit :: Profunctor p => p :-> Cotambara (Copastro p)

counit :: Profunctor p => Copastro (Cotambara p) :-> p

Profunctor (Copastro p)

Defined in Data.Profunctor.Strong

Methods

dimap :: (a -> b) -> (c -> d) -> Copastro p b c -> Copastro p a d

lmap :: (a -> b) -> Copastro p b c -> Copastro p a c

rmap :: (b -> c) -> Copastro p a b -> Copastro p a c

(#.) :: Coercible c b => q b c -> Copastro p a b -> Copastro p a c

(.#) :: Coercible b a => Copastro p b c -> q a b -> Copastro p a c

Costrong (Copastro p)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Copastro p (a, d) (b, d) -> Copastro p a b

unsecond :: Copastro p (d, a) (d, b) -> Copastro p a b

ProfunctorFunctor Copastro

Defined in Data.Profunctor.Strong

Methods

promap :: Profunctor p => (p :-> q) -> Copastro p :-> Copastro q